Correlated random walks in heterogeneous landscapes: Derivation, homogenization, and invasion fronts

Frithjof Lutscher; Thomas Hillen; Frithjof Lutscher; Thomas Hillen

doi:10.3934/math.2021518

AIMS Mathematics

2021, Volume 6, Issue 8: 8920-8948. doi: 10.3934/math.2021518

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Research article Special Issues

Correlated random walks in heterogeneous landscapes: Derivation, homogenization, and invasion fronts

Frithjof Lutscher ^{1
,
,},
Thomas Hillen ²

1.
Department of Mathematics and Statistics, and Department of Biology, University of Ottawa, Ottawa, ON, K1N6N5, Canada
2.
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada

Received: 01 August 2020 Accepted: 01 June 2021 Published: 15 June 2021
MSC : 35B27, 35L40, 92D40

Many models for the movement of particles and individuals are based on the diffusion equation, which, in turn, can be derived from an uncorrelated random walk or a position-jump process. In those models, individuals have a location but no well-defined velocity. An alternative, and sometimes more accurate, model is based on a correlated random walk or a velocity-jump process, where individuals have a well defined location and velocity. The latter approach leads to hyperbolic equations for the density of individuals, rather than parabolic equations that result from the diffusion process. Almost all previous work on correlated random walks considers a homogeneous landscape, whereas diffusion models for uncorrelated walks have been extended to spatially varying environments. In this work, we first derive the equations for a correlated random walk in a one-dimensional spatially varying environment with either smooth variation or piecewise constant variation. Then we show how to derive the so-called parabolic limit from the resulting hyperbolic equations. We develop homogenization theory for the hyperbolic equations, and show that taking the parabolic limit and homogenization are commuting actions. We illustrate our results with two examples from ecology: the persistence and spread of a population in a patchy heterogeneous landscape.
- correlated random walk,
- hyperbolic differential equation,
- multi-scale,
- homogenization,
- heterogeneous landscape,
- population persistence,
- population spread rate
Citation: Frithjof Lutscher, Thomas Hillen. Correlated random walks in heterogeneous landscapes: Derivation, homogenization, and invasion fronts[J]. AIMS Mathematics, 2021, 6(8): 8920-8948. doi: 10.3934/math.2021518

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Abstract

Many models for the movement of particles and individuals are based on the diffusion equation, which, in turn, can be derived from an uncorrelated random walk or a position-jump process. In those models, individuals have a location but no well-defined velocity. An alternative, and sometimes more accurate, model is based on a correlated random walk or a velocity-jump process, where individuals have a well defined location and velocity. The latter approach leads to hyperbolic equations for the density of individuals, rather than parabolic equations that result from the diffusion process. Almost all previous work on correlated random walks considers a homogeneous landscape, whereas diffusion models for uncorrelated walks have been extended to spatially varying environments. In this work, we first derive the equations for a correlated random walk in a one-dimensional spatially varying environment with either smooth variation or piecewise constant variation. Then we show how to derive the so-called parabolic limit from the resulting hyperbolic equations. We develop homogenization theory for the hyperbolic equations, and show that taking the parabolic limit and homogenization are commuting actions. We illustrate our results with two examples from ecology: the persistence and spread of a population in a patchy heterogeneous landscape.

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